Article Figures & Data
Figures
- FIGURE 1.
Slice of Zubal phantom. Inverse linear gray scale is used for comparison with original phantom.
- FIGURE 2.
Support region (hatched area) of 2-dimensional autoregressive model. Current predicted pixel (gray area) is in middle of region.
- FIGURE 3.
Iteratively filtered image (A) summed with filtered error term image (B) to get final image (C). Images are individually scaled to their own maximum. Count level is intermediate.
- FIGURE 4.
Low- (A), intermediate- (C), and high-count (E) phantom images corrupted by Poisson noise. Noise was removed using best adaptive autoregressive model (B, D, and F). Images are individually scaled to their own maximum.
- FIGURE 5.
One-pixel-thick profile curves drawn through Zubal phantoms, noise-corrupted Zubal phantoms, and phantoms filtered using best adaptive autoregressive model, at level shown by phantom. Count levels are low (A), intermediate (B), and high (C). AAR = adaptive autoregressive model.
- FIGURE 6.
Low-, intermediate-, and high-count phantom images filtered with 3 × 3 mean filter (A, C, and E) and 3 × 3 median filter (B, D, and F). Images are individually scaled to their own maximum.
- FIGURE 7.
Noisy projection image of skeletal SPECT (A), and projection image with noise removed (B). Adaptive autoregressive model is used, with prediction region of 4 orthogonal neighbors of predicted pixel and block size of 5 × 5 pixels.
- FIGURE 8.
Adaptive autoregressive filter was applied to projection data, and transversal slices were reconstructed with iterative method. Shown are reformatted coronal (A) and sagittal (B) slices of skeletal SPECT image. Adaptive autoregressive model is used, with prediction region of 4 orthogonal neighbors of predicted pixel and block size of 5 × 5 pixels. No additional noise removal methods were applied.
Tables
- TABLE 1
Mean Squared Errors for Different Combinations of Prediction Region and Block Size When Degree of Overlap Is 50%
Prediction region Low count level Intermediate count level High count level Block size 3 × 3* 3 × 3 5 × 5 3 × 3* 3 × 3 5 × 5 3 × 3* 3 × 3 5 × 5 4 × 4 2.09 1.73 — 8.83 3.17 — 19.12 6.29 — 5 × 5 1.31 1.23 — 3.93 3.11 — 13.32 10.50 — 6 × 6 1.30 1.19 8.41 3.95 3.29 19.17 13.72 11.70 110.07 7 × 7 1.26 1.04 1.28 3.93 3.06 3.65 13.60 10.76 14.12 8 × 8 1.28 1.07 1.17 3.97 3.21 3.53 13.99 11.31 11.92 9 × 9 1.28 1.06 1.06 3.95 3.12 3.16 14.00 11.17 11.25 10 × 10 1.25 1.09 1.15 3.95 3.32 3.38 14.01 11.78 11.81 11 × 11 1.25 1.06 1.06 3.94 3.28 3.22 14.05 11.80 11.56 12 × 12 1.26 1.09 1.08 3.98 3.46 3.48 14.17 12.30 12.51 3 × 3* = prediction region of 4 orthogonal neighbors of predicted pixel; — = modeling not possible because prediction region ≥ block size.
Block sizes and other prediction regions are squares in pixels.
- TABLE 2
Mean Squared Errors for Different Combinations of Prediction Region and Block Size When Degree of Overlap Is 75%
Prediction region Low count level Intermediate count level High count level Block size 3 × 3* 3 × 3 5 × 5 3 × 3* 3 × 3 5 × 5 3 × 3* 3 × 3 5 × 5 5 × 5 1.26 1.08 — 3.85 2.95 — 13.25 10.11 — 6 × 6 1.24 1.01 3.28 3.80 2.89 9.02 13.29 10.11 42.9 7 × 7 1.23 1.00 1.07 3.83 2.94 3.11 13.43 10.39 10.72 8 × 8 1.25 1.02 1.04 3.87 3.06 3.10 13.66 10.87 10.91 9 × 9 1.24 1.02 1.03 3.88 3.08 3.07 13.75 11.02 11.01 10 × 10 1.25 1.04 1.02 3.94 3.22 3.13 13.99 11.55 11.19 11 × 11 1.25 1.05 1.02 3.96 3.26 3.17 14.07 11.72 11.38 12 × 12 1.22 1.07 1.03 3.99 3.38 3.23 14.21 12.15 11.74 3 × 3* = prediction region of 4 orthogonal neighbors of predicted pixel; — = modeling not possible because prediction region = block size.
Block sizes and other prediction regions are squares in pixels.
- TABLE 3
Effect of Iteration on Mean Squared Errors for Different Combinations of Prediction Region and Block Size
Prediction region Low count level Intermediate count level High count level Block size n 3 × 3* 3 × 3 5 × 5 3 × 3* 3 × 3 5 × 5 3 × 3* 3 × 3 5 × 5 5 × 5 0 1.26 1.08 — 3.85 2.95 — 13.25 10.11 — 1 1.05 0.97 — 3.12 2.99 — 10.47 10.62 — 2 1.32 1.05 — 4.33 3.39 — 15.36 12.11 — 6 × 6 0 1.24 1.01 3.28 3.80 2.89 9.02 13.29 10.11 42.9 1 1.01 0.92 3.51 3.00 2.91 26.05 10.28 10.44 40.69 2 1.27 1.02 4.56 4.19 3.31 82.05 15.21 12.20 49.32 7 × 7 0 1.23 1.00 1.07 3.83 2.94 3.11 13.43 10.39 10.72 1 1.00 0.91 1.00 3.00 2.97 3.26 10.35 10.50 11.29 2 1.26 1.02 1.08 4.20 3.41 3.61 15.37 12.50 12.63 8 × 8 0 1.25 1.02 1.04 3.87 3.06 3.10 13.66 10.87 10.91 1 1.01 0.93 0.94 3.06 3.05 3.20 10.63 10.85 11.06 2 1.27 1.04 1.02 4.26 3.54 3.53 15.57 13.00 12.52 9 × 9 0 1.24 1.02 1.03 3.88 3.08 3.07 13.75 11.02 11.01 1 0.99 0.91 0.96 3.04 3.02 3.14 10.67 10.69 11.02 2 1.25 1.04 1.04 4.23 3.56 3.50 15.75 13.07 12.32 10 × 10 0 1.25 1.04 1.02 3.94 3.22 3.13 13.99 11.55 11.19 1 1.01 0.94 0.94 3.12 3.14 3.15 11.08 11.39 10.83 2 1.27 1.07 1.03 4.33 3.75 3.49 16.24 13.81 12.41 11 × 11 0 1.25 1.05 1.02 3.96 3.26 3.17 14.07 11.72 11.38 1 1.00 0.94 0.96 3.12 3.14 3.16 11.15 11.36 10.90 2 1.27 1.08 1.04 4.34 3.80 3.52 16.37 14.01 12.53 12 × 12 0 1.22 1.07 1.03 3.99 3.38 3.23 14.21 12.15 11.74 1 1.01 0.96 0.97 3.15 3.23 3.18 11.38 11.88 11.07 2 1.27 1.10 1.05 4.39 3.90 3.54 16.69 14.48 12.73 n = number of iterations; 3 × 3* = prediction region of 4 orthogonal neighbors of predicted pixel; — = modeling not possible because prediction region = block size.
Block sizes and other prediction regions are squares in pixels.
Prediction region Low count level Intermediate count level High count level Block size n 3 × 3* 3 × 3 5 × 5 3 × 3* 3 × 3 5 × 5 3 × 3* 3 × 3 5 × 5 5 × 5 0 1.05 0.97 — 3.12 2.99 — 10.47 10.62 — 1 0.95 0.91 — 2.79 2.57 — 9.25 9.20 — 2 0.85 0.88 — 2.23 2.32 — 7.12 7.87 — 6 × 6 0 1.01 0.92 3.51 3.00 2.91 26.05 10.28 10.44 40.69 1 0.92 0.86 6.13 2.71 2.49 28.05 9.19 8.85 58.47 2 0.84 0.83 5.15 2.19 2.23 143.63 7.25 7.61 40.92 7 × 7 0 1.00 0.91 1.00 3.00 2.97 3.26 10.35 10.50 11.29 1 0.91 0.85 0.94 2.73 2.56 2.72 9.31 8.95 9.15 2 0.85 0.81 0.93 2.22 2.26 2.50 7.45 7.75 8.25 8 × 8 0 1.01 0.93 0.94 3.06 3.05 3.20 10.63 10.85 11.06 1 0.92 0.86 0.89 2.80 2.63 2.69 9.61 9.31 9.20 2 0.85 0.82 0.87 2.27 2.33 2.47 7.75 8.10 8.16 9 × 9 0 0.99 0.91 0.96 3.04 3.02 3.14 10.67 10.69 11.02 1 0.91 0.85 0.89 2.79 2.61 2.67 9.66 9.21 9.26 2 0.85 0.82 0.85 2.29 2.34 2.42 7.87 8.08 8.34 10 × 10 0 1.01 0.94 0.94 3.12 3.14 3.15 11.08 11.39 10.83 1 0.93 0.87 0.88 2.87 2.73 2.71 10.04 9.83 9.16 2 0.86 0.82 0.85 2.33 2.43 2.47 8.14 8.66 8.19 11 × 11 0 1.00 0.94 0.96 3.12 3.14 3.16 11.15 11.36 10.90 1 0.92 0.87 0.89 2.87 2.72 2.72 10.02 9.84 9.27 2 0.86 0.83 0.85 2.34 2.43 2.47 8.24 8.67 8.31 12 × 12 0 1.01 0.96 0.97 3.15 3.23 3.18 11.38 11.88 11.07 1 0.93 0.89 0.90 2.91 2.81 2.75 10.28 10.37 9.52 2 0.87 0.84 0.85 2.38 2.52 2.51 8.38 9.09 8.49 n = number of summed error term images before filtering; 3 × 3* = prediction region of 4 orthogonal neighbors of predicted pixel; — = modeling not possible because prediction region = block size.
Block sizes and other prediction regions are squares in pixels.
- TABLE 5
Total Counts and Mean Squared Errors of Poisson Noise–Corrupted Images and Images After Removal of Noise Using Best Adaptive Autoregressive Model, 3 × 3 Mean Filter, and 3 × 3 Median Filter
Low count level Middle count level High count level Parameter Total counts MSE Total counts MSE Total counts MSE Poisson noise 28596 1.73 54210 3.17 108780 6.29 Adaptive autoregressive model 29240 0.81 54477 2.19 110950 7.12 Mean filter 28596 0.83 54210 2.71 108780 9.85 Median filter 27015 0.82 52483 2.21 106471 6.54 MSE = mean squared error.
